As is known, devices have been proposed that are able to reveal the quantum behaviour of Nature. In particular, devices are known that enable detection of the quantum behaviour of particles such as, for example, photons. The operation of these devices is based on single or multiple photon state processing and finds application in particularly advanced sectors such as, for example, quantum computing, quantum cryptography, quantum communications and even random number generation.
In particular, the so-called Hong-Ou-Mandel interferometer, also known as the HOM interferometer, is also known, an example of which is shown in FIG. 1, where it is indicated with reference numeral 1.
In detail, the HOM interferometer 1 comprises an optical source 2, a crystal 4, a delay line 6, a polarization phase-shifter 8, first and second mirrors 10, 12 and an optical beam splitter 14. The HOM interferometer 1 also comprises a beam stopper 16.
In greater detail, the optical source 2 is a coherent type of source, such as a laser source for example.
The crystal 4 is an optically non-linear crystal such as, for example, a non-centrosymmetric crystal (for example, a crystal of barium borate, BBO), and is aligned with the optical source 2.
The delay line 6 is of the optical type and is formed, for example, by a so-called optical trombone. In use, when it is crossed by a photon, the delay line 6 delays it for a predetermined time.
Whereas with regard to the polarization phase-shifter 8, this is formed by a birefringent crystal, which delays photons having different polarizations differently, namely by introducing controlled phase shifting, typically not greater than the wavelength of the photons. For example, the polarization phase-shifter 8 can be voltage controlled.
In greater detail, the crystal 4 defines, together with the delay line 6, the polarization phase-shifter 8 and the first mirror 10, a first optical path 20. The crystal 4 also defines, together with the second mirror 12, a second optical path 22. The delay line 6 is able to alter the optical length of the first optical path 20 with respect to the second optical path 22.
The optical beam splitter 14 is of the so-called 50/50 type and has a first and a second input and a first and a second output. From the classical point of view, given an incident signal on any input of the first and second inputs, two signals will be generated on the first and second outputs of the optical beam splitter 14 having power half that of the incident signal.
The first and second optical paths 20, 22 are optically and respectively connected to the first and second inputs of the optical beam splitter 14.
Operationally, the optical source 2 is able to emit electromagnetic pulses formed of photons at the same frequency, which are commonly referred to as pump photons; these electromagnetic pulses, and therefore the pump photons, impinge on the crystal 4.
In particular, assuming that a pump photon impinge on the crystal 4, due to the phenomenon of spontaneous parametric down-conversion (SPDC), the crystal 4 can generate a pair of converted photons, one of which propagates along the first optical path 20, while the other propagates along the second optical path 22. Instead, in the case where there is no spontaneous parametric down-conversion, the pump photon passes through the crystal 4 and is absorbed by the beam stopper 16, which, for this purpose, is arranged in front of the crystal 4, with which it is aligned.
In the case where two converted photons are generated, both of them propagate until reaching a respective input of the optical beam splitter 14. In this regard, generally, but not necessarily, the optical beam splitter 14 is formed by a pair of prisms suitable for receiving electromagnetic signals that propagate in free space, such that the term “input” implies a corresponding direction of propagation of an electromagnetic signal or a photon that impinges on the optical beam splitter 14, while the term “output” implies a corresponding direction of propagation of an electromagnetic signal or a photon that moves away from the optical beam splitter 14.
The term “first converted photon” is used to indicate the photon of the pair of converted photons that propagates along the first optical path 20; this photon arrives at the first input of the optical beam splitter 14 after having passed through the delay line 6 and the polarization phase-shifter 8, and also after being reflected by the first mirror 10. Furthermore, when the first converted photon impinges on the optical beam splitter 14, it can, alternatively and with the same probability, pass through the optical beam splitter 14, leaving from the first output of the optical beam splitter 14, or be reflected by the optical beam splitter 14, leaving from the second output of the optical beam splitter 14.
Similarly, the term “second converted photon” is used to indicate the photon of the pair of converted photons that propagates along the second optical path 22; this photon arrives at the second input of the optical beam splitter 14 after being reflected by the second mirror 12. Furthermore, when the second converted photon impinges on the optical beam splitter 14, it can, alternatively and with the same probability, pass through the optical beam splitter 14, leaving from the second output of the optical beam splitter 14, or can be reflected by the optical beam splitter 14, leaving from the first output of the optical beam splitter 14.
In other words, in the case where the first converted photon passes through the optical beam splitter 14, it does not change its direction of propagation. Conversely, when the first converted photon is reflected by the optical beam splitter 14, the direction of propagation is changed. Furthermore, the crystal 4, the first and second mirrors 10, 12 and the optical beam splitter 14 are arranged such that, in case of reflection, the first converted photon propagates with a direction of propagation equal to the direction of propagation with which the second converted photon impinges on the optical beam splitter 14.
Completely symmetrical considerations can be made regarding the second converted photon. In fact, in the case where second converted photon passes through the optical beam splitter 14, it does not change its direction of propagation. Conversely, when the second converted photon is reflected by the optical beam splitter 14, the direction of propagation is changed. Furthermore, the crystal 4, the first and second mirrors 10, 12 and the optical beam splitter 14 are arranged such that, in case of reflection, the second converted photon propagates with a direction of propagation equal to the direction of propagation with which the first converted photon impinges on the optical beam splitter 14.
Thus, in the case where first converted photon passes through the optical beam splitter 14 and the second converted photon is reflected by the optical beam splitter 14, they subsequently propagate along a same direction of propagation (first output of the optical beam splitter). Similarly, in the case where the first converted photon is reflected by the optical beam splitter 14 and the second photon passes through the optical beam splitter 14, they subsequently propagates along a same direction of propagation (second output of the optical beam splitter).
In order to detect the quantum behaviour of the photons, it is possible to prepare a first and a second photodetector 30, 32, both of the single photon type, i.e. able to detect individual photons. For example, the first and second photodetectors 30, 32 could be Geiger-mode avalanche photodiodes, also known as single-photon avalanche photodiodes (SPAD).
The first photodetector 30 is placed to detect the first converted photon if it passes through the optical beam splitter 14 and the second converted photon if it is reflected by the optical beam splitter 14. Similarly, the second photodetector 32 is placed to detect the first converted photon if it is reflected by the optical beam splitter 14 and the second converted photon if it passes through the optical beam splitter 14.
There are therefore four different possible scenarios for output from the optical beam splitter 14, detectable by using the first and second photodetectors 30, 32 and known as Feynman paths:
a) both the first and second converted photons pass through the optical beam splitter 14;
b) both the first and the second converted photon are reflected by the optical beam splitter 14;
c) the first converted photon passes through the optical beam splitter 14, while the second converted photon is reflected by the optical beam splitter 14; and
d) the second converted photon passes through the optical beam splitter 14, while the first converted photon is reflected by the optical beam splitter 14.
From the quantum viewpoint, assuming (temporarily) for simplicity that the first and second converted photons have the same polarization, and assuming that the delay line 6 and the polarization phase-shifter 8 are such as to compensate possible differences in length between the first and second optical paths 20, 22, in a manner such that the first and second converted photons impinge on the optical beam splitter 14 at the same time, contrary to that foreseen by classical optics, it is found that scenarios a) a b) can never occur. In other words, the first and second converted photons always find themselves on the same output of the optical beam splitter 14. In particular, the probability that the first and second converted photons are on the first output of the optical beam splitter 14 is 0.5, and the probability that they are on the second output of the optical beam splitter 14 is 0.5.
Thus, the first and second converted photons cannot both be reflected or both be transmitted, and therefore both be detected by the same photodetector, whether this be the first photodetector 30 or the second photodetector 32. By monitoring the first and second photodetectors 30, 32, a total lack of coincidence between their measurements is therefore verified. This is due to destructive interference between different alternatives; in the case in point, between scenario a) and scenario b).
In general, the described phenomenon is usually referred to as coalescence, or that the first and second converted photons are coalescent.
More precisely, the phenomenon of coalescence of a pair of photons occurs when, as in the example described, it is not possible to distinguish between scenario a) and scenario b), for example, by measuring different arrival times of the first and second converted photons at the first and second photodetectors 30, 32.
According to one interpretation of the phenomenon of the coalescence of two photons, the optical beam splitter 14 is a linear device capable of discriminating between symmetric states, namely between invariant states with respect to particle exchange, and asymmetric states, also known as antisymmetric states.
In detail, a state of a system with two photons having the same polarization is symmetric, precisely because it is invariant with respect to the exchange of the two photons. In fact, when two photons impinge on two different inputs of an optical beam splitter of the 50/50 type, they are propagated together in output from the optical beam splitter, along the same direction.
By using the so-called Dirac notation, the input state to the optical beam splitter 14 is normally indicated |11|12, where the subscripts “1” and “2” refer to the first and the second optical path 20, 22 respectively. Regarding this, it is normal to still refer to the first output of the optical beam splitter 14 as the first optical path 20, and still refer to the second output of the optical beam splitter 14 as the second optical path 22, so that it is possible to express the output state of the optical beam splitter 14 as 1/√{square root over (2)}(|21,02−|01,22), from which it can be inferred that there is only one antisymmetric state with two coalescent photons in output from the HOM interferometer 1.
More in detail, it is possible to remove the simplifying assumption in which the first and second converted photons have the same polarization, something that, in effect, does not occur in the case where, as shown in FIG. 1, they are generated by the crystal 4 through type-II spontaneous parametric down-conversion. In fact, in this case, the pump photon is annihilated and the first and second converted photons are generated with orthogonal polarizations; in other words, if one of the first and second converted photons propagates along the first optical path 20 with horizontal polarization, the other photon propagates on the second optical path 22 with vertical polarization, or vice versa.
In particular, in the case of so-called type-II spontaneous parametric down-conversion, it is found that the first and second converted photons are orthogonally polarized and satisfy the so-called phase-matching conditions, i.e. the conditions of conservation of energy and linear moment.
It thus occurs that one of the first and second converted photons is polarized in a direction parallel to the optical axis of the crystal 4, also known as the extraordinary direction, while the other is polarized in a direction perpendicular to the optical axis of the crystal 4, also known as the ordinary direction. Furthermore, due to conservation of the linear moment, as shown in FIG. 2, the first and second converted photons are respectively emitted along a first and a second emission cone 34, 36, respectively corresponding to the extraordinary direction and the ordinary direction.
More in detail, assuming the degenerate case, namely the case where the first and second converted photons both have frequencies equal to half of the frequency of the pump photon, the first and second converted photons emerge from the crystal 4 forming a precise angle, for example, respectively equal to ±3°, with respect to the direction of propagation of the pump photon, as shown in FIGS. 2 and 3 for example.
The first and second emission cones 34, 36 intersect along a first and a second line 38, 40, along which it is therefore possible to detect both the first and the second converted photons. In other words, along the first and second lines 38, 40 it is possible to detect polarized photons both along the extraordinary direction and along the ordinary direction. Therefore, the HOM interferometer 1 is such that the first and second optical paths 20, 22 respectively lie along the first and second lines 38, 40, which undergo obvious changes following the interaction of the first and second emission cones 34, 36 with the components of the HOM interferometer 1, such as, for example, the first and second mirrors 10, 12.
That having been said, using the term “extraordinary photon” to indicate the photon, of the first and second converted photons, polarized parallel to the optical axis of the crystal 4, and using the term “ordinary photon” to indicate the photon, of the first and second converted photons, polarized perpendicularly to the optical axis of the crystal 4, it is possible to discriminate between state |e1|o2 and state |o1|e2. In other words, it is possible to distinguish a scenario in which the extraordinary photon (also known as the “signal”) and the ordinary photon (also known as the “idler”) respectively propagate along the first and second optical path 20, 22 (state |e1|o2) from a scenario in which the extraordinary photon and the ordinary photon respectively propagate along the second and the first optical path 22, 20 (state |o1|e2). This is due to the fact that the extraordinary photon and the ordinary photon propagate in the crystal 4 with different group velocities, and so the corresponding emissions are temporally distinguishable.
In practice, the input state to the optical beam splitter 14 is still coherent and can be expressed as:
                                                                                        e                〉                            1                        ⁢                                                          o                〉                            2                                +                                    ⅇ                              ⅈ                ⁢                                                                  ⁢                φ                                      ⁢                                                          o                〉                            1                        ⁢                                                          e                〉                            2                                                2                                    (        1        )            where φ is a function of the phase shift introduced by the polarization phase-shifter 8.
In practice, by means of the delay line 6 and the polarization phase-shifter 8, it is possible to control the input state to the optical beam splitter 14, as well as the superposition of the wave function of the two-photon system on the optical beam splitter 14.
For example, when φ=0, the input state is symmetric and so coalescence of the photons occurs. Here, φ is such that the condition φ=0 is obtained when the delay introduced by the delay line 6 is sufficient to compensate the difference in optical length between the first and second optical paths 20, 22, and when the polarization phase-shifter 8 is inactive.
Conversely, in the case where φ=π, the following is obtained:
                                                                                        e                〉                            1                        ⁢                                                          o                〉                            2                                -                                                                    o                〉                            1                        ⁢                                                          e                〉                            2                                                2                                    (        2        )            namely an antisymmetric state is obtained, also known as a singlet state. In particular, relation (2) can also be expressed as:
                                                                                      1                                  e                  ⁢                                                                          ⁢                  1                                            〉                        ⁢                                                        1                                  2                  ⁢                  o                                            〉                                -                                                                  1                                  1                  ⁢                  o                                            〉                        ⁢                                                        1                                  2                  ⁢                  e                                            〉                                                2                                    (        3        )            where subscripts “1” and “2” still refer to the first and second optical paths 20, 22 and subscripts “e” and “o” refer to the extraordinary photon and the ordinary photon.
It can then be checked, both mathematically and experimentally, that the singlet state does not change, i.e. that the state on output from the optical beam splitter 14 still takes form (2), equivalent to form (3).
Therefore, the coalescence of the extraordinary photon and the ordinary photon does not occur in output from the optical beam splitter 14. Conversely, anti-coalescence occurs, since the extraordinary photon and the ordinary photon are always present on different outputs of the optical beam splitter 14. In other words, the optical beam splitter 14 implements a projection of the input state in the symmetric and antisymmetric subspaces, this projection also being known as Bell measurement.
By way of example, anti-coalescence can be detected, as shown in FIG. 4, by using a third and a fourth photodetectors 42, 44, and two further optical beam splitters, which are referred to as the first and second measurement splitters 46, 48. In particular, the first and second measurement splitters 46, 48 are polarizing optical beam splitters, each of which is able to let one of either the extraordinary photon or the ordinary photon pass through and to reflect the other one, so as to spatially separate the extraordinary photon and the ordinary photon. For example, the first measurement splitter 46 can be arranged on the first output of the optical beam splitter 14 in a manner such that, in the case where the extraordinary photon or the ordinary photon emerge from the first output of the optical beam splitter 14, they are respectively directed towards the first photodetector 30 and the third photodetector 42.
Similarly, the second measurement splitter 48 can be arranged on the second output of the optical beam splitter 14 in a manner such that, in the case where extraordinary photon or the ordinary photon emerge from the second output of the optical beam splitter 14, they are respectively directed towards the second photodetector 32 and the fourth photodetector 44.
In practice, by counting the readings of the first, second, third and fourth photodetectors 30, 32, 42, 44, it is possible to determine measurements related to the so-called probabilities 1e1o, 1e2o, 2e1o and 2e2o, i.e. the probabilities that:                both the extraordinary photon and the ordinary photon are on the first output of the optical beam splitter 14;        the extraordinary photon and the ordinary photon are, respectively, on the first and the second output of the optical beam splitter 14;        the extraordinary photon and the ordinary photon are, respectively, on the second and the first output of the optical beam splitter 14; and        both the extraordinary photon and the ordinary photon are on the second output of the optical beam splitter 14.        
More in detail, the description concerning the input state to the optical beam splitter 14 can be rendered mathematically more accurate in relation to the physical phenomenon. In fact, by considering the longitudinal components of the electromagnetic fields associated with the photons, the emission state from the crystal 4 can be expressed as:
                                        ψ          〉                =                              C                          2                                ⁢                                    ∫                              -                L                            0                        ⁢                                          ⅆ                z                            ⁢                                                ∫                  0                                      +                    ∞                                                  ⁢                                                      ⅆ                                          v                      p                                                        ⁢                                                            E                      p                                              (                        +                        )                                                              ⁡                                          (                                              v                        p                                            )                                                        ⁢                                      ⅇ                                          ⅈ                      ⁢                                                                                          ⁢                                              v                        p                                            ⁢                      Λ                      ⁢                                                                                          ⁢                      z                                                        ⁢                                                            ∫                                              -                        ∞                                                                    +                        ∞                                                              ⁢                                                                  ⅆ                        v                                            ⁢                                                                                          ⁢                                              ⅇ                                                                              -                            ⅈ                                                    ⁢                                                                                                          ⁢                          Dvz                                                                    ×                      ×                                              [                                                                                                                                                                              a                                  ^                                                                                                  1                                  ⁢                                  e                                                                †                                                            ⁡                                                              (                                                                  v                                  +                                                                                                                                                    v                                        p                                                                            +                                                                              Ω                                        p                                                                                                              2                                                                                                  )                                                                                      ⁢                                                                                                                            a                                  ^                                                                                                  2                                  ⁢                                  o                                                                †                                                            ⁡                                                              (                                                                                                      -                                    v                                                                    +                                                                                                                                                    v                                        p                                                                            +                                                                              Ω                                        p                                                                                                              2                                                                                                  )                                                                                                              -                                                                                                                                                      a                                  ^                                                                                                  2                                  ⁢                                  e                                                                †                                                            ⁡                                                              (                                                                  v                                  +                                                                                                                                                    v                                        p                                                                            +                                                                              Ω                                        p                                                                                                              2                                                                                                  )                                                                                      ⁢                                                                                                                            a                                  ^                                                                                                  1                                  ⁢                                  o                                                                †                                                            ⁡                                                              (                                                                                                      -                                    v                                                                    +                                                                                                                                                    v                                        p                                                                            +                                                                              Ω                                        p                                                                                                              2                                                                                                  )                                                                                                                                    ]                                            ⁢                                                                      0                        〉                                                                                                                                                    (        4        )            where C is a constant that depends on the power of the optical source 2, the interaction volume of the pump photons with the crystal 4 and the second-order, non-linear, effective tensor of the crystal 4. In addition, Ep(+)(υp) is the spectral distribution of the pump, namely of the electromagnetic radiation emitted by the optical source 2, Ωp is the central pump frequency and L is the length of the crystal 4, measured along the propagation direction of the pump. Moreover, {circumflex over (α)}1e† and {circumflex over (α)}2e† are the creation operators related to the extraordinary photon and to the first and second optical paths 20, 22, respectively; {circumflex over (α)}1o† and {circumflex over (α)}2o† are the creation operators related to the ordinary photon and to the first and second optical paths 20, 22, respectively. The following also hold:
                              Λ          =                                    1                              u                p                                      -                                          1                2                            ⁢                              (                                                      1                                          u                      e                                                        +                                      1                                          u                      o                                                                      )                                                    ,                                  ⁢        and                            (        5        )                                          D          =                      (                                          1                                  u                                      e                    ⁢                                                                                                                            -                              1                                  u                  o                                                      )                          ,                            (        6        )            where up, ue and uo are the group velocities in the crystal 4 of the pump, the extraordinary photon and the ordinary photon, respectively.
In practice, equation (4) permits revealing observance of the energy conservation condition in the generation process of the extraordinary photon and the ordinary photon.
The output state from the optical beam splitter 14 can thus be expressed as:
                                        ψ          〉                =                                            C              ⁢                                                          ⁢                              ⅇ                                                      -                                          ⅈ                      ⁡                                              (                                                                              Ω                            p                                                    2                                                )                                                                              ⁢                  ζ                                                                    2              ⁢                              2                                              ⁢                                    ∫                              -                L                            0                        ⁢                                          ⅆ                z                            ⁢                                                ∫                  0                                      +                    ∞                                                  ⁢                                                      ⅆ                                          v                      p                                                        ⁢                                                            E                      p                                              (                        +                        )                                                              ⁡                                          (                                              v                        p                                            )                                                        ⁢                                      ⅇ                                          ⅈ                      ⁢                                                                                          ⁢                                                                        v                          p                                                ⁡                                                  (                                                                                    Λ                              ⁢                                                                                                                          ⁢                              z                                                        -                                                          ζ                              2                                                                                )                                                                                                      ⁢                                                            ∫                                              -                        ∞                                                                    +                        ∞                                                              ⁢                                                                  ⅆ                        v                                            ⁢                                                                                          ⁢                                              ⅇ                                                                              -                            ⅈ                                                    ⁢                                                                                                          ⁢                          D                          ⁢                                                                                                          ⁢                          vz                                                                    ×                                              {                                                                                                                                                                                                                  [                                                                                                                                                                                                                                                                                                                                                                                                                                                                  α                                                      ^                                                                                                                                                              1                                                      ⁢                                                      e                                                                                                        †                                                                                                    ⁡                                                                                                      (                                                                                                          v                                                      +                                                                                                                                                                                                                                    v                                                            p                                                                                                                    +                                                                                                                      Ω                                                            p                                                                                                                                                                          2                                                                                                                                                              )                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          a                                                        ^                                                                                                                                                                    1                                                        ⁢                                                        o                                                                                                            †                                                                                                        ⁡                                                                                                          (                                                                                                                                                                        -                                                          v                                                                                                                +                                                                                                                                                                                                                                            v                                                              p                                                                                                                        +                                                                                                                          Ω                                                              p                                                                                                                                                                                2                                                                                                                                                                    )                                                                                                                                                        -                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                a                                                      ^                                                                                                                                                              2                                                      ⁢                                                      e                                                                                                        †                                                                                                    ⁡                                                                                                      (                                                                                                          v                                                      +                                                                                                                                                                                                                                    v                                                            p                                                                                                                    +                                                                                                                      Ω                                                            p                                                                                                                                                                          2                                                                                                                                                              )                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  a                                                      ^                                                                                                                                                              2                                                      ⁢                                                      o                                                                                                        †                                                                                                    ⁡                                                                                                      (                                                                                                                                                                  -                                                        v                                                                                                            +                                                                                                                                                                                                                                    v                                                            p                                                                                                                    +                                                                                                                      Ω                                                            p                                                                                                                                                                          2                                                                                                                                                              )                                                                                                                                                                                                                                                                                                                                                                                            ]                                                                    ⁢                                                                      (                                                                                                                  ⅇ                                                                                  ⅈ                                          ⁢                                                                                                                                                                          ⁢                                          v                                          ⁢                                                                                                                                                                          ⁢                                          ζ                                                                                                                    ±                                                                              ⅇ                                                                                                                              -                                            ⅈ                                                                                    ⁢                                                                                                                                                                          ⁢                                          v                                          ⁢                                                                                                                                                                          ⁢                                          ζ                                                                                                                                                      )                                                                                                  +                                                                                                                                                                                                                                          [                                                                                                                                                                                                                                                                                                                                                                                                                                              a                                                    ^                                                                                            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           +                                                                                                                  Ω                                                          p                                                                                                                                                                    2                                                                                                                                                        )                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    a                                                      ^                                                                                                                                                              2                                                      ⁢                                                      o                                                                                                        †                                                                                                    ⁡                                                                                                      (                                                                                                                                                                  -                                                        v                                                                                                            +                                                                                                                                                                                                                                    v                                                            p                                                                                                                    +                                                                                                                      Ω                                                            p                                                                                                                                                                          2                                                                                                                                                              )                                                                                                                                                  -                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  a                                                    ^                                                                                                                                                        2                                                    ⁢                                                    e                                                                                                    †                                                                                                ⁡                                                                                                  (                                                                                                      v                                                    +                                                                                                                                                                                                                            v                                                          p                                                                                                                +                                                                                                                  Ω                                                          p                                                                                                                                                                    2                                                                                                                                                        )                                                                                                                                                                                                                                                                                                                                                                                                                                                                              a                                                    ^                                                                                                                                                        1                                                    ⁢                                                    o                                                                                                    †                                                                                                ⁡                                                                                                  (                                                                                                                                                            -                                                      v                                                                                                        +                                                                                                                                                                                                                            v                                                          p                                                                                                                +                                                                                                                  Ω                                                          p                                                                                                                                                                    2                                                                                                                                                        )                                                                                                                                                                                                                                                                                                                                                                          ]                                                                ⁢                                                                  (                                                                                                            ⅇ                                                                              ⅈ                                        ⁢                                                                                                                                                                  ⁢                                        v                                        ⁢                                                                                                                                                                  ⁢                                        ζ                                                                                                              ∓                                                                          ⅇ                                                                                                                        -                                          ⅈ                                                                                ⁢                                                                                                                                                                  ⁢                                        v                                        ⁢                                                                                                                                                                  ⁢                                        ζ                                                                                                                                              )                                                                                                                                                                    }                                            ⁢                                                                      0                        〉                                                                                                                                                    (        7        )            where symbol “±” is intended to be “+” in the case when φ=o (symmetric input state) and “−” when φ=π (antisymmetric input state); furthermore, in equation (7), ζ is a time that is a function of the delay introduced by the delay line 6.
The above-mentioned (normalized) probabilities 1e1o, 1e2o, 2e1o and 2e2o, hereinafter respectively indicated as HOM1e1o, HOM1e2o, HOM2e1o and HOM2e2o, can thus be expressed as:
                                                                        HOM                                  1                  ⁢                  e                  ⁢                                                                          ⁢                  1                  ⁢                  o                                            ⁡                              (                ς                )                                      =                                          1                4                            [                              1                ±                                                      Tri                    ⁡                                          (                                              ς                        DL                                            )                                                        ⁢                                      ⅇ                                                                  -                        2                                            ⁢                                              σ                        p                        2                                            ⁢                                                                        Λ                          2                                                                          D                                                      2                            ⁢                                                                                                                                                                                    ⁢                                              ς                        2                                                                                                        ]                                ,                                          ⁢                                                    HOM                                  1                  ⁢                  e                  ⁢                                                                          ⁢                  2                  ⁢                  o                                            ⁡                              (                ς                )                                      =                                          1                4                            [                              1                ∓                                                      Tri                    ⁡                                          (                                              ς                        DL                                            )                                                        ⁢                                      ⅇ                                                                  -                        2                                            ⁢                                                                                          ⁢                                              σ                        p                        2                                            ⁢                                                                        Λ                          2                                                                          D                                                      2                            ⁢                                                                                                                                                                                    ⁢                                              ς                        2                                                                                                        ]                                ,                                          ⁢                                                    HOM                                  2                  ⁢                  e                  ⁢                                                                          ⁢                  1                  ⁢                  o                                            ⁡                              (                ς                )                                      =                                          1                4                            [                              1                ∓                                                      Tri                    ⁡                                          (                                              ς                        DL                                            )                                                        ⁢                                      ⅇ                                                                  -                        2                                            ⁢                                              σ                        p                        2                                            ⁢                                                                        Λ                          2                                                                          D                          2                                                                    ⁢                                              ς                        2                                                                                                        ]                                      ⁢                                  ⁢                                                            HOM                                  2                  ⁢                  e                  ⁢                                                                          ⁢                  2                  ⁢                  o                                            ⁡                              (                ς                )                                      =                                          1                4                            [                              1                ±                                                      Tri                    ⁡                                          (                                              ς                        DL                                            )                                                        ⁢                                      ⅇ                                                                  -                        2                                            ⁢                                              σ                        p                        2                                            ⁢                                                                        Λ                          2                                                                          D                          2                                                                    ⁢                                              ς                        2                                                                                                        ]                                ,                                    (        8        )            where the following holds:
                              Tri          ⁡                      (            x            )                          =                  {                                                    0                                                                                  for                    ⁢                                                                                  ⁢                                                                x                                                                              >                                      1                    2                                                                                                                        (                                      1                    -                                          2                      ⁢                                                                      x                                                                                                      )                                                                                                  for                    ⁢                                                                                  ⁢                                                                x                                                                              ≤                                      1                    2                                                                                                          (        9        )            
As shown in FIG. 5, in the case where the input state is symmetric, it is found that the probabilities (more precisely, the corresponding probability density functions) HOM1e1o and HOM2e2o have a maximum of 0.5 when ζ=0. In other words, coalescence occurs, since the first and second converted photons are on the same output of the optical beam splitter 14, which, with a probability of 0.5, is the first output or, with the same probability, is the second output of the optical beam splitter 14.
As shown in FIG. 6, in the case where the input state is antisymmetric, it is found that the probabilities HOM1e1o and HOM2e2o are null when ζ=0. In other words, anti-coalescence occurs, since the extraordinary photon and the ordinary photon are never present on the same output of the optical beam splitter 14.
From a practical standpoint, the HOM interferometer 1 has been used, among other things, in measuring the temporal uncertainty associated with the emission of a pair of photons by spontaneous parametric down-conversion, which is typically less than a picosecond. Still by way of example, the HOM interferometer 1 has also been used to test the Bell inequalities, to measure the tunneling time of the photons, to perform quantum teleportation, and even to build quantum logic gates, to clone quantum states and to implement quantum key distribution schemes.
Even though the HOM interferometer 1 enables generating the coalescence of two photons and consequently lends itself to various applications based on the quantum characteristics of matter, it does not allow its outputs to be controlled, i.e. the probabilities with which the two coalescent photons appear on the first or on the second output of the optical beam splitter 14.